You are an engineer in charge of designing the dimensions of a box-like building. The base is rectangular in shape with width being twice as large as length. (Therefore so is the ceiling.) The volume is to be 1944000 m3. Local bylaws stipulate that the building must be no higher than 30 m. Suppose the walls cost twice as much per m2 as the ceiling, and suppose the floor (i.e.base) costs nothing. Find the dimensions of the building that would minimize the cost.
To find the dimensions of the building that minimize the cost, we need to formulate an equation for the cost in terms of the dimensions of the building.
Let's say the length of the base of the building is "x". Therefore, the width of the base is "2x" according to the given information.
The height of the building is limited to 30 m, so the height of the building is "h = 30".
The volume of the building is given as 1944000 m^3, which is equal to the product of the base area (length times width) and the height. Therefore, we can write the equation as:
Volume = base area * height
1944000 = x * 2x * 30
1944000 = 60x^2
x^2 = 1944000 / 60
x^2 = 32400
x = √32400
x = 180 m
Now that we know the length of the base (x = 180 m), we can find the width of the base, which is twice the length:
Width of the base = 2x
Width = 2 * 180 = 360 m
So, the dimensions of the base of the building are 180 m x 360 m.
To find the cost, we need to find the area of the walls and the area of the ceiling. The cost of the walls is twice the cost per m^2 of the ceiling. Let's assume the cost per m^2 of the ceiling is "$C". Therefore, the cost per m^2 of the walls will be "$2C".
The area of the walls can be calculated by subtracting the base area from the total surface area, which is the sum of both the walls and the ceiling. The surface area of the ceiling is the area of the base.
Total Surface Area = 2 * (area of walls) + area of base
2 * (area of walls) = Total Surface Area - area of base
2 * (area of walls) = 2 * (length * width) - (length * width)
Now, substituting the values we found earlier:
2 * (area of walls) = 2 * (180 * 360) - (180 * 360)
2 * (area of walls) = 2 * 64800 - 64800
2 * (area of walls) = 2 * 64800 - 64800
2 * (area of walls) = 64800
Dividing both sides of the equation by 2 gives us:
area of walls = 32400
The cost of the walls is twice the cost per m^2 of the ceiling, so the cost of the walls can be calculated as:
Cost of walls = 2C * (area of walls) = 2C * 32400
The cost of the ceiling is simply the cost per m^2 of the ceiling multiplied by the area of the base:
Cost of ceiling = C * (length * width) = C * (180 * 360)
Therefore, the total cost can be expressed as:
Total Cost = Cost of walls + Cost of ceiling
Total Cost = 2C * 32400 + C * (180 * 360)
To minimize the cost, we need to find the value of C that minimizes the total cost. This can be done by taking the derivative of the total cost equation with respect to C and setting it to zero. Then solve for C to find the minimum cost.
Differentiating the total cost equation with respect to C:
d(Total Cost)/dC = 2 * 32400 + 180 * 360 = 0
Simplifying the equation:
64800 + 64800 = 0
129600 = 0
This equation is not possible since it implies that 129600 is equal to 0, which is not true. Therefore, there is no minimum cost since the derivative does not equal zero.
In conclusion, the dimensions of the building that would minimize the cost cannot be determined.